# What is the angle between <-4,8,0>  and < 1,0,-2>?

Mar 11, 2018

${78}^{o} \setminus$ to nearest degree

#### Explanation:

The angle, $\theta$, between vectors $\boldsymbol{\underline{a}}$ and $\boldsymbol{\underline{b}}$ is given by:

$\boldsymbol{\underline{a}} \cdot \boldsymbol{\underline{b}} = | \boldsymbol{\underline{a}} | \setminus | \boldsymbol{\underline{b}} | \setminus \cos \theta$

So if we define:

$\boldsymbol{\underline{a}} = \left\langle- 4 , 8 , 0\right\rangle$
$\boldsymbol{\underline{b}} = \left\langle1 , 0 , - 2\right\rangle$

Then we calculate the scalar (or dot product):

$\boldsymbol{\underline{a}} \cdot \boldsymbol{\underline{b}} = \left\langle- 4 , 8 , 0\right\rangle \cdot \left\langle1 , 0 , - 2\right\rangle$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = \left(- 4\right) \left(1\right) + \left(8\right) \left(0\right) + \left(0\right) \left(- 2\right)$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = - 4$

And we calculate the norms:

$| \boldsymbol{\underline{a}} | = \sqrt{{\left(- 4\right)}^{2} + {8}^{2} + {0}^{2}}$
$\setminus \setminus \setminus \setminus \setminus = \sqrt{16 + 64 + 0}$
$\setminus \setminus \setminus \setminus \setminus = \sqrt{80}$

$| \boldsymbol{\underline{b}} | = \sqrt{{1}^{2} + {0}^{2} + {\left(- 2\right)}^{2}}$
$\setminus \setminus \setminus \setminus \setminus = \sqrt{1 + 0 + 4}$
$\setminus \setminus \setminus \setminus \setminus = \sqrt{5}$

So using $\boldsymbol{\underline{a}} \cdot \boldsymbol{\underline{b}} = | \boldsymbol{\underline{a}} | \setminus | \boldsymbol{\underline{b}} | \setminus \cos \theta$ we can write:

$- 4 = \sqrt{80} \sqrt{5} \setminus \cos \theta$

$\therefore \cos \theta = - \frac{4}{\sqrt{400}}$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = - \frac{1}{5}$

$\therefore \theta = \arccos \left(- \frac{1}{5}\right)$
$\setminus \setminus \setminus \setminus \setminus \setminus = 78.46304 \ldots$
$\setminus \setminus \setminus \setminus \setminus \setminus = {78}^{o} \setminus$ to nearest degree